MHF 3202 Reasoning and Proof in Mathematics
Basic Information
Instructor: Vincent Vatter
Office: Little Hall 406
Email: vatter@ufl.edu
Credit hours: 3
Grading scheme: Letter Grade
Class meetings: MWF 7th period (1:55–2:45) in LIT 233
Office hours: Monday 5th period (11:45–12:35) in LIT 406, Tuesday 9th period (4:05–4:55) in LIT 225, or by appointment
Required textbook: None (lecture notes will be supplied)
Recommended text: Richard Hammack, Book of Proof (available free online).
Course Description
Fundamentals of logic, set theory, and proof techniques. Topics include propositional and predicate logic, set operations, functions, relations, mathematical induction, and proof strategies. Emphasis on analyzing, constructing, writing, and communicating rigorous mathematical arguments with clarity and precision.
Prerequisite: MAC 2312 (Calculus II) with a minimum grade of C.
Course Objectives
By the end of this course, students will be able to:
- Construct direct and indirect proofs, including with mathematical induction.
- Construct counterexamples to disprove false statements.
- Analyze the logical structure of statements.
- Apply propositional and predicate logic to mathematical arguments.
- Use set theory to define and manipulate mathematical objects.
- Understand and apply relations and functions in proof-based contexts.
- Detect and correct logical errors in mathematical reasoning.
- Communicate mathematical concepts and arguments clearly and rigorously.
Assessment
- Exams (60%): Three in-class, non-comprehensive exams on Wednesday, February 4, Wednesday, March 4, and Wednesday, April 15. Each exam is worth 20% of the final grade. Exams are closed book and closed notes, held in the classroom. Make-up exams require prior agreement or documentation of an excused absence.
- Homework (40%): For homework, you are encouraged to refer to your notes and to consult with your classmates and AI (see AI policy below).
- Late policy: Late assignments will only be accepted by prior agreement or in the case of an excused absence.
- Regrade requests: If you have a disagreement with the grading of one of your solutions, please submit a written request for reconsideration within one month.
- No final exam.
Expectations and Grading Rubric
Work submitted for a grade in this course will be graded in a rigorous fashion and should be prepared with a good deal of thought and care.
Most of the work required in this course will consist of writing proofs. For a proof worth 10 points, scores will be based on the following guidelines:
0 points: The work contains no original steps toward a correct solution. This includes work that simply consists of relevant definitions or theorems without interpretation.
3 points: The work contains some original steps toward a correct solution but does not contain a workable outline of the full solution. This grade is also used if the student has misunderstood the question or made an unwarranted simplifying assumption that makes the problem trivial.
6 points: The work contains an outline of a correct solution and several steps toward this solution. However, the writing may be unclear, or there may be holes in the argument.
8 points: The work resembles a full, complete proof, but it has some deficiencies. These may include incomplete sentences, abbreviating words with logical symbols such as those for “for all” or “implies”, imprecise definitions, or overlooking trivial cases.
10 points: The work consists of a full, complete proof and is reasonably well written in complete sentences, without logical symbols. There may be minor typos or clumsy writing that could be improved, but no important steps of the solution are omitted or incorrect.
Grade Distribution
Grades are assigned based on total percentage earned, with no rounding, according to the table below.
| A | 93.00–100.00 | A- | 90.00–92.99 | ||
| B+ | 87.00–89.99 | B | 83.00–86.99 | B- | 80.00–82.99 |
| C+ | 77.00–79.99 | C | 73.00–76.99 | C- | 70.00–72.99 |
| D+ | 67.00–69.99 | D | 63.00–66.99 | D- | 60.00–62.99 |
| E | 0.00–59.99 |
Schedule of Topics
| Week | Topic | Notes |
|---|---|---|
| Week 1 | Integers, parity, implications, direct proof | Jan 12–16 |
| Week 2 | Proof by cases, contrapositives, converses, biconditionals | Jan 19–23 (MLK Mon) |
| Week 3 | Proof by contradiction, divisibility | Jan 26–30 |
| Week 4 | Divisors, division algorithm, Bézout’s identity | Exam 1 (Wed 2/4) |
| Week 5 | Euclid’s lemma, fundamental theorem of arithmetic | Feb 9–13 |
| Week 6 | Infinitely many primes, rationality and irrationality | Feb 16–20 |
| Week 7 | Introduction to sets and set operations | Feb 23–27 |
| Week 8 | Set operation properties, Cartesian products, power sets | Exam 2 (Wed 3/4) |
| Week 9 | Relations and their properties, functions | Mar 9–13 |
| Spring Break | Mar 16–20 | |
| Week 10 | Composition of functions, injectivity, surjectivity, bijectivity | Mar 23–27 |
| Week 11 | Mathematical induction, strong induction | Mar 30–Apr 3 |
| Week 12 | Fibonacci numbers, permutations, binomial coefficients | Apr 6–10 |
| Week 13 | Binomial identities, Hilbert’s hotel, countability | Exam 3 (Wed 4/15) |
| Week 14 | Additional topics | Apr 20–22 |
Materials and Supplies Fee
N/A
AI Policy
You may consult any resources when working on problem sets, including generative AI tools (e.g., ChatGPT, Claude, Gemini), computer algebra systems (CASes), textbooks, online materials, classmates, and the instructor. However, the solutions you submit must be your own writing, and you must understand and be able to explain your solutions. Copying solutions from any external source, whether an AI tool, a textbook, or a classmate, will be considered a violation of academic integrity.
Use of external resources during exams is not permitted.
University Policies and Resources
This course complies with all UF academic policies. For information on those policies and for resources for students, please see https://syllabus.ufl.edu/syllabus-policy/uf-syllabus-policy-links/.